Disturbance and Error Term - Econometrics

All You Need to Know About Disturbance in Econometrics-

a.    For BLUE variables should be iid.

b.    IID is important in the classical form of “Central Limit Theorem”.

c.     Disturbance “u” is the error in the true model and “ε” is of the estimated model.

d.    In case of time series disturbance is considered to have dynamic structure with zero mean and  is defined as an innovation or white noise.

e.    Intercept guarantees the mean of residuals to be zero and with zero intercept all the regression variables are forced to be zero resulting in biased estimates.

f.      The distribution is potential distribution i.e. generated before the sample is generated. The probability of u reaching a given positive or negative value will be same in all observations. This is called homoscedasticity. Weighted Least square for linear models and logarithmic regression for non-linear models can be used to overcome heteroskedasticity.

g.    Coefficient properties depends on the disturbance term

h.    Standard error of regression coefficient is calculated on the assumption that distribution of the disturbance (u) is homoscedastic.

i.       Coefficient value is Normal distributed if each disturbance term in each observation is Normally distributed.

j.       Measurement error and Simultaneous equation bias occurs when disturbance and independent variable are not independent.

k.    For normally distributed and consistent estimates for large samples variables should be covariance stationary.

l.       R-squared always increases with the addition of new variable, even if the additional variable is uncorrelated with the dependent variable. 

m. Coefficients can be significant despite of low R² as R² do not consider intercept.

n.    It is possible to have variables that are dependent but uncorrelated, since correlation only measures linear dependence. A wonderful thing about normally distributed RV’s is that they are a convenient special case: if they are uncorrelated, they are also independent.

o.    Expected value rule can be applied in non-stochastic regressors but not for stochastic regressors.

p.    Correlation is equal to covariance for standardized random variables.

q.    Generally, all macroeconomic variables are likely to be endogenous and as such none can act as a valid instrument i.e. IV.

The solution is the inclusion of lagged variables as an explanatory variable. Provided u_t is not serially correlated.

r.      Reduced form model parameters are nonlinear functions of the structural model parameters.

s.     True value of coefficient is estimated value + linear combination of disturbance term in all the observations in the sample.

-         U_t must be distributed independently to at

-         Since a_t depends on all observations of independent variable (X) u_t must be independent to all the observations of X.

For cross sectional data (b) is seldom an issue. Since the observations are generated randomly, there is no no reason to suppose that disturbance in one is not independent of other observation of regressors.

---